Tensor Visualization Chap. 7 April 2, 2013 April 4, Jie Zhang Copyright

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1 Tenor iualization Chap. 7 April, 3 April 4, 3 Jie Zhang Copright CDS 3 Spring, 3

2 Outline 7.. Principle Component Anali (PCA 7.. iualizing Component 7.3. iualizing Scalar PCA Information 7.4. iualizing ector PCA Information 7.5. Tenor Glph 7.6. Fiber Tracking 7.7. Hpertreamline

3 Tenor z,, ( ˆ ˆ ˆ : ector or or k j i z z f : Scalar zz z z z z Tenor : T

4 Tenor Curvature tenor: curvature of a 3-D urface Stre and train tenor in mechanical engineering Diffuion tenor e.g., water in tiue diffue in different peed in different direction e.g, in human brain tiue, diffuion i tronger in the direction of the neural fiber; meaured b DT-MRI (diffuion tenor magnetic reonance imaging icoit tenor in fluid

5 Eample Curvature of a 3-D urface i a tenor. Curvature i different along different direction

6 Eample A curved urface in 3- D z f (, The Gauian Function : e ( f ( Gradient (in - D pace f f, f ( f, f Eq.5 pace Normal Direction (in 3 - D pace n n ( f, f, (f,f, Eq.4

7 Eample : along Curvature : along Curvature f f Curvature calculation eq 3.4 ee matri (alo Heian : in co vector direction unit arbituar an i : along Curvature f f f f h h h h H H H f f T

8 Eample Quetion: for the following Gauian function that define a 3-D urface, ( Calculate it normal direction at point (,, ( Calculate the curvature along α=, 45 and 9 degree at point (, (3 Calculate the gradient of the correponding - D function at point (, z e (

9 Eample, (,, (, in co At,, 4 (, 4 4, 4 ( (,,,, ( ( H f α α f f f f H f f n e f T

10 iualizing Component Data: (3 X 3 Diffuion tenor data from DT-MRI can Show onl one lice of the 3-D data Nine Component.

11 Principal Component Anali (PCA A tenor ha principal direction, e.g., along which the curvature are etremal (maimum or minimum One could prove,if H Curvature along i etremal : eigenvector λ: eigenvalue

12 PCA (normalized and orthogonal find then, Find ( ( det( Determinant - det matri. identif i I where ( h h h h I H h h h h I H I (H I H

13 PCA: eample Quetion: calculate the eigenvalue and eigenvector of the following tenor H 3 4

14 PCA: eample (continued 5 7 (4 (3,4, 3 det det( I H

15 PCA: eample (continued ,4, 3 H For Similarl

16 (April, 3 Stop Here

17 April 4, 3

18 Review: PCA (normalized and orthogonal find then, Find Determinant det matri. identif i I where ( I (H I H S HS h h h h H

19 PCA If we order the eigenvalue in decreaing order: 3 e : major eigenvecto r e e 3 : medium eigenvecto r : minor eigenvecto r In cae of curvature: e the direction of maimal curvature e the direction of minimum curvature e 3 : the direction of urface normal

20 PCA

21 Average Diagonal Entrie h 3 ( h h h 33 Average diffuion

22 iualizing Aniotrop the mean diffuivit i ( ( 3 Fa Aniotrop Fractional 3 3 i i the mean diffuivit i ( ( 3 Fa Relative Aniotrop 3 3 i i meaure pherical ; 3 c planar meaure ; - ( c linear meaure ; - c Certaint : p 3 L

23 iualizing Aniotrop

24 iualizing ector PCA Show major eigenvector direction uing color coding Shaded color phere: R: horizontal (X G: vertical (Y B: depth (Z

25 Tenor Glph Ellipoid Tenor Glph In triangle glph pace Three corner: Linear, Planar, Spherical

26 Tenor Glph DT-MRI Ellipoid Tenor Glph Color: direction on haded phere

27 Fiber Tracking Similar to vector treamline Tracing along the major eigenvector Start from a eed region

28 Hpertreamline Similar to tream tube The tream line i not convolved with a circular cro ection But an elliptic cro ection, whoe ae are along the direction of the medium and minor eigenvector.

29 Hpertreamline

30 MATLAB: PCA Quetion: calculate the eigenvalue and eigenvector of the following tenor H 3 4

31 MATLAB: PCA >> H =[3, ; 4] ; %define the matri >>d=eig(h; >>[v, d] = eig(h %find the eigenvalue %v: eigenvector, d: eigenvalue % To prove: H * v = d * v %Firt ector: v(:, >>H * v(:, >>d(, * v(:, %the two hould be the ame

32 MATLAB: matri calculation Quetion: for the following Gauian function that define a 3-D urface, ( Find the Heian matri at point (, ( Calculate the curvature along α=, 45 and 9 degree at point (, z e (

33 , (,, (, in co At,, 4 (, 4 4, 4 ( (,,,, ( ( H f α α f f f f H f f n e f T MATLAB: matri calculation

34 MATLAB: matri calculation >> H =[-, ;, -] ; %define the Heian matri >>alpha = ; %define the direction: deg >> = [co(alpha,in(alpha] ; %direction vector >>curv=*h*tranpoe(; %find the curvature %Angle 45 degree >>alpha = 45.*pi/8 >> = [co(alpha,in(alpha] >>curv=*h*tranpoe(;

35 End of Chap. 7

Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization. CSC 7443: Scientific Information Visualization Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its